Vietoris–Rips filtration explained

In topological data analysis, the Vietoris–Rips filtration (sometimes shortened to "Rips filtration") is the collection of nested Vietoris–Rips complexes on a metric space created by taking the sequence of Vietoris–Rips complexes over an increasing scale parameter. Often, the Vietoris–Rips filtration is used to create a discrete, simplicial model on point cloud data embedded in an ambient metric space.^[1] The Vietoris–Rips filtration is a multiscale extension of the Vietoris–Rips complex that enables researchers to detect and track the persistence of topological features, over a range of parameters, by way of computing the persistent homology of the entire filtration.^{[2] [3] [4]} It is named after Leopold Vietoris and Eliyahu Rips.

Definition

The Vietoris–Rips filtration is the nested collection of Vietoris–Rips complexes indexed by an increasing scale parameter. The Vietoris–Rips complex is a classical construction in mathematics that dates back to a 1927 paper^[5] of Leopold Vietoris, though it was independently considered by Eliyahu Rips in the study of hyperbolic groups, as noted by Mikhail Gromov in the 1980s.^[6] The conjoined name "Vietoris–Rips" is due to Jean-Claude Hausmann.^[7] Given a metric space

X and a scale parameter (sometimes called the threshold or distance parameter) r \in [ 0 , inf ty) , the Vietoris–Rips complex (with respect to r ) is defined as VR _{r(X)} = \{ \emptyset ≠ S \subseteq X \mid S finite ; \operatorname{diam} S \leq r \} , where \operatorname{diam} S is the diameter, i.e. the maximum distance of points lying in S .^[8] Observe that if r \leq s \in [ 0 , inf ty) , there is a simplicial inclusion map VR _{r(X)} \hookrightarrow VR _{s(X)} . The Vietoris–Rips filtration is the nested collection of complexes VR _{r(X)} :

$$\backslash mathbf(X)\; =\; \backslash \_$$

Notes and References

1. Book: Chazal . Frédéric . Interleaved Filtrations: Theory and Applications in Point Cloud Data Analysis . 2013 . Geometric Science of Information . 8085 . 587–592 . Nielsen . Frank . 2023-04-05 . Berlin, Heidelberg . Springer Berlin Heidelberg . 10.1007/978-3-642-40020-9_65 . 978-3-642-40019-3 . Oudot . Steve Y. . 8701910 . Barbaresco . Frédéric.
2. Dey . Tamal K. . Shi . Dayu . Wang . Yusu . 2019-01-30 . SimBa: An Efficient Tool for Approximating Rips-filtration Persistence via Simplicial Batch Collapse . ACM Journal of Experimental Algorithmics . 24 . 1.5:1–1.5:16 . 10.1145/3284360 . 216028146 . 1084-6654. free .
3. Book: Oudot . Steve Y. . Sheehy . Donald R. . Proceedings of the twenty-ninth annual symposium on Computational geometry . Zigzag zoology . 2013-06-17 . https://doi.org/10.1145/2462356.2462371 . SoCG '13 . New York, NY, USA . Association for Computing Machinery . 387–396 . 10.1145/2462356.2462371 . 978-1-4503-2031-3. 485245 .
4. Book: Lee . Hyekyoung . Chung . Moo K. . Kang . Hyejin . Kim . Bung-Nyun . Lee . Dong Soo . 2011 IEEE International Symposium on Biomedical Imaging: From Nano to Macro . Discriminative persistent homology of brain networks . 2011 . https://ieeexplore.ieee.org/document/5872535/;jsessionid=EZNTPzFc08Tk2MdVbttcE6ev01Li3fXvqhHf7iHVLJT4_mI8ZaOj!71486806 . 841–844 . 10.1109/ISBI.2011.5872535. 978-1-4244-4127-3 . 12511452 .
5. Vietoris . L. . 1927-12-01 . Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen . Mathematische Annalen . de . 97 . 1 . 454–472 . 10.1007/BF01447877 . 121172198 . 1432-1807.
6. Book: Gromov, M. . Hyperbolic Groups . 1987 . Essays in Group Theory . Mathematical Sciences Research Institute Publications . 8 . 75–263 . Gersten . S. M. . 2023-04-05 . New York, NY . Springer New York . 10.1007/978-1-4613-9586-7_3 . 978-1-4613-9588-1.
7. Reitberger, Heinrich (2002), "Leopold Vietoris (1891–2002)", Notices of the American Mathematical Society, 49 (20).
8. Book: Bauer . Ulrich . Roll . Fabian . 2022 . Goaoc . Xavier . Kerber . Michael . Gromov Hyperbolicity, Geodesic Defect, and Apparent Pairs in Vietoris–Rips Filtrations . 38th International Symposium on Computational Geometry (SoCG 2022) . Leibniz International Proceedings in Informatics (LIPIcs) . Dagstuhl, Germany . Schloss Dagstuhl – Leibniz-Zentrum für Informatik . 224 . 15:1–15:15 . 10.4230/LIPIcs.SoCG.2022.15 . free . 978-3-95977-227-3. 245124031 .